4.1. Application
The model was applied on precipitation and streamflow data detailed in
Section 3 with summary characteristics in
Table 3 and
Table 4. Details of the application on an example gauging station are as follows for each step of the model:
(a) Nondimensionalization: First, streamflow data were nondimensionalized by using Equation (1); i.e., streamflow discharge at each month is divided by the long-term average streamflow discharge of the gauging station. In order to demonstrate how the model is applied at this step, an example is shown by using data from gauging station D20A019, which had an observation period extending over 10 years from 1963 to 1972 (
Table 5).
(b) Normalization: In this step, the non-normal data were converted into normal distribution. Within the 8664 station-month streamflow data in
Table 4, 31 station-months only were recorded as zero, which were excluded from the normalization. The average and standard deviation of the non-zero monthly streamflow discharge records were calculated after which
θ = 0.166 was obtained by the trial-and-error to make the data set normal. It is seen from the histogram in
Figure 4 and the P–P plot in
Figure 5 that the data were normalized properly. The ratio of
was calculated as 1.01, which simply shows the goodness-of-fit of the normal distribution to the data after the transformation. The transformed data were further checked by the
test that approved the normalization. All these checks and tests showed that the normalization was achieved. In
Table 6, the nondimensional monthly streamflow data are given for the example gauging station (D20A019).
(c) Calculation of normal quantiles: The standard normal variable was given for different exceedance probabilities
in
Table 7. Given that the mean and standard deviation are available, the transformed streamflow of any exceedance probability
could be calculated by Equation (3),
= 1.05 for
= 25% exceedance probability as an example.
(d) Back transformation of nondimensional quantiles: Equation (4) was used for the back transformation of the nondimensional streamflow discharge data (
;
Table 7);
= 1.31 for
= 25% as an example (
Table 8,
Figure 6).
(e) Calculation of the mean streamflow: The dimensional quantiles of FDC can be calculated by using the empirical regression equation, which is developed between the mean streamflow and watershed characteristics among which the drainage area and slope of each watershed were taken into account together with precipitation. Following equations were obtained to alternate each other in calculating the average streamflow discharge of the Ceyhan River basin.
In calculating the average streamflow discharge at the calibration stage, Equation (12) was found the best based on the determination coefficient (
R2) and the root mean square error (
RMSE) while Equation (10) performed the best based on the mean absolute error (
MAE;
Table 9). For the validation stage, however, Equation (10) performed better than all others although the performance metrics did not differ from each other considerably. The similar performance of the alternative equations could bring the concept of the equifinality to mind [
43], yet, this is not an issue to be considered here. Instead, the simplest regression equation in terms of parameterization was found the most successful in reproducing the average streamflow discharge. Not only because of this reason but also due to its parsimonious structure in terms of parameters and the number of inputs, Equation (10) was preferred in this study. The fitted curve based on Equation (10) is plotted in
Figure 7. The calculated streamflow discharges of each gauging station were compared with their counterparts as in
Figure 8 from which a good fit was observed.
(f) Calculation of dimensional quantiles: Any dimensional discharge (
) corresponding to the exceedance probability
can be calculated by Equation (6) using the dimensionless discharge corresponding to the same exceedance probability (
) and the average streamflow (
) calculated from Equation (10). This was applied on the validation gauging stations of the Ceyhan River basin to obtain their monthly FDCs as in
Figure 9.
4.2. Discussion
In this study, in order to prove that the model is applicable to ungauged river basins, a monthly streamflow data set composed of 42 hydrometric stations was chosen from the Ceyhan River basin in southern Turkey as detailed above. FDCs of the model were compared to the observed FDCs of the gauged hydrometric stations, which were particularly chosen to show the similarity between the modeled and observed FDCs. Once the similarity is satisfied, the model can be proposed to develop FDC at any un-gauged site. This was the way in how the model was validated in this study.
Figure 9 shows FDCs for the validation gauging stations (14 in total) for which performance metrics were obtained as in
Table 10. When FDCs in
Figure 9 were taken into account together with the performance criteria in
Table 10, the general result that came out was that the methodology performed quite well for the validation gauging stations. This was clear particularly due to the performance criteria in
Table 10, which were almost all either very good, good or adequate except for the
VE in one station (D20A063) and
BiasFLV in four (D20A044, D20A056, D20A059 and D20A068). The total number of inadequate cases was five out of 98 when the number of gauging stations and number of performance metrics were considered together. It is important to note also that
BiasFLV is a performance metric for the lower end of the FDC where flow is quite low. Therefore, even if
BiasFLV shows an inadequate performance in these particular stations, the deviation from the observation in terms of the streamflow volume is not substantial but ignorable.
When results were analyzed, FDCs were generally considered successful as a whole and at once. However, the part-by-part analysis of FDCs is quite beneficial particularly for the lower tail with low flows. It is then possible to get more detail of the FDC. This was the case here particularly when the lower tail of the FDC did not seem to be satisfactory. The deviation was clear for exceedance probabilities higher than 70% in gauging stations D20A005, D20A007, D20A054 and D20A055. However, when
BiasFLV, the performance metric applicable to the lower tail of the FDC was checked in
Table 10, it was seen that it stayed in the range of “very good”, which means the visual deviation in these particular gauging stations did not necessarily mean a large-volume error. On the other hand, although the lower-tail demonstrated an “inadequate” performance in terms of the
BiasFLV for gauging stations D20A044, D20A056, D20A059 and D20A068, the visual disagreement in
Figure 9 was not as clear as in the above-mentioned gauging stations. The inadequate performance for the lower tail of the flow duration curve in these particular stations might be ignored; because the difference between the observed and the modeled streamflow volumes was not that substantial as the discharge was quite small. In gauging station E20A009, which is one of the two gauging stations where streamflow was intermittent, the performance metrics were found very satisfactory for the FDC. Nevertheless, in the second intermittent streamflow gauging station (D20A044), a deviation was seen between the observation and the model as this particular station had a very low streamflow discharge. This discussion can also be based on the fact that the deviation between the lower tails of the modeled and observed FDCs increased with the degree of intermittency of the river due to higher between-year variability of the FDCs in drier rivers than the wet rivers [
1]. It should also be noted that logarithmic and probabilistic scales were used in the vertical and horizontal axes in
Figure 9, respectively, that might exaggerate the deviation visually.
When the generalization of the results was needed, it could be seen that the modeled FDCs were underestimated for some gauging stations while they were overestimated for others. It is also seen that FDCs could behave differently in different parts; i.e., an FDC that underestimates the upper tail of the FDC might overestimate the lower tail or vice versa. This could be connected with the lumped behavior of the empirical regression equation in which the calibration gauging stations were all combined and forced to be represented by one free variable and two parameters only. Only one free variable existed in the regression equation as the basin drainage area and precipitation were used in their multiplicative form. Yet, results of the regression equation were considered acceptable.
The basic aim of this study was to develop a flow duration curve model for ungauged sites at the monthly time step. As a pre-requisite, the model needs a regression equation that links the long-term mean streamflow of the river basin to the multiplicative form of its drainage area and precipitation. The former can simply be delineated and calculated by modern geographic information systems technology while the latter is generally available or it can be obtained from areal distribution maps of precipitation. With the existence and relatively easy availability of drainage area and precipitation, the FDC model is applicable to any ungauged site within a river basin, conditional that a regression equation calculating streamflow from the drainage area and precipitation is given. The performance of the regression model is at the utmost importance. It is expected that the smaller the river basin, the better the performance of the regression equation. In this sense, the model can generally be used for smaller-size ungauged sub-basins surrounded by larger gauged river basins. However, the model could still be considered a solution to the problem of the FDC development for ungauged basins, one of the most important unsolved problems in hydrology [
23,
44].